Question

Minimum Payments for Credit Cards Holders of credit cards issued by banks, department stores, oil companies, and so on, receive bills each month that state minimum amounts that must be paid by a certain due date. The minimum due depends on the total amount owed. One such credit card company uses the following rules: For a bill of less than $$\$ 10$$ the entire amount is due. For a bill of at least $$\$ 10$$ but less than $$\$ 500$$, the minimum due is $$\$ 10$$. A minimum of $$\$ 30$$ is due on a bill of at least $$\$ 500$$ but less than $$\$ 1000$$, a minimum of $$\$ 50$$ is due on a bill of at least $$\$ 1000$$ but less than $$\$ 1500,$$ and a minimum of $$\$ 70$$ is due on bills of $$\$ 1500$$ or more. Find the function \(f\) that describes the minimum payment due on a bill of \(x\) dollars. Graph \(f\)

Short answer

The function is a piecewise function defined as:\[ f(x) = \begin{cases} x & \text{if } x < 10 10 & \text{if } 10 \leq x < 500 30 & \text{if } 500 \leq x < 1000 50 & \text{if } 1000 \leq x < 1500 70 & \text{if } x \geq 1500 \end{cases} \]

Step-by-step solution

Understanding the Problem

Given the rules, we need to determine the minimum payment due based on different ranges of a bill amount, denoted as x dollars. We will construct a function f(x) that describes these rules.

Define the Piecewise Function

We need to consider different cases based on the value of x:- For x < 10, the entire amount (x) is due.- For 10 <= x < 500, the minimum payment is \(10.- For 500 <= x < 1000, the minimum payment is \)30.- For 1000 <= x < 1500, the minimum payment is \(50.- For x >= 1500, the minimum payment is \)70.

Construct the Piecewise Function Expression

Based on the cases, we can write the function as follows:\[ f(x) = \begin{cases} x & \text{if } x < 10 \ 10 & \text{if } 10 \leq x < 500 \ 30 & \text{if } 500 \leq x < 1000 \ 50 & \text{if } 1000 \leq x < 1500 \ 70 & \text{if } x \geq 1500 \end{cases} \]

Graph the Function

Create a graph where the x-axis represents the bill amount and the y-axis represents the minimum payment due. Plot the function according to the constructed piecewise expression, ensuring that each segment matches the corresponding rule defined above.

Key concepts

Minimum Payments

Learning about minimum payments is essential for managing credit card bills efficiently.
Minimum payments are the smallest amount you must pay towards your bill each month to keep your account in good standing.
Not paying this minimum can lead to late fees and damage your credit score.

In the provided exercise, the minimum payment varies based on the total amount owed:

• If the bill is less than \(10, the entire amount is due.
• For bills from \)10 to under \(500, a fixed minimum of \)10 is due.
• From \(500 to under \)1000, the minimum payment is \(30.
• From \)1000 to under \(1500, the minimum payment is \)50.
• If the bill is \(1500 or more, a minimum of \)70 is due.

Understanding these rules helps in planning payments better.

Graphing Functions

Graphing a piecewise function means plotting segments based on different cases.
In the exercise, the graph helps visualize the relationship between the bill amount and the minimum payment due.
To graph this specific function:

• The x-axis represents the bill amount (x dollars).
• The y-axis represents the minimum payment due (f(x)).
• Plot each segment according to the rules given in the piecewise function.

For example, for a bill less than \(10, plot y = x (a line passing through the origin). For bills from \)10 to under $500, plot y = 10 (a horizontal line). Do this similarly for all the segments. Each piece will be either a line or a constant.
This graphical representation can make it easier to understand the payment structure.

Understanding Piecewise Functions

Piecewise functions are special because they have different expressions for different intervals of the independent variable.
In this exercise, the variable x (bill amount) determines which rule applies for the minimum payment.
Understanding piecewise functions involves:

• Identifying the different cases or intervals.
• Defining a different expression for each interval.
• Combining these expressions into a single piecewise function.

Piecewise functions are useful in real-life scenarios where rules or behaviors change based on conditions, like tax brackets, shipping costs, and of course, credit card minimum payments.

Piecewise Function Definition

A piecewise function is defined by multiple sub-functions, each of which applies to a certain interval of the independent variable's domain.
The piecewise function from this exercise can be defined as:
oindent \[ f(x) = \begin{cases} x & \text{if } x < 10 \ 10 & \text{if } 10 \leq x < 500 \ 30 & \text{if } 500 \leq x < 1000 \ 50 & \text{if } 1000 \leq x < 1500 \ 70 & \text{if } x \geq 1500 \end{cases} \]
This notation means that the function f(x) takes different forms depending on the value of x.
Each condition specifies a different segment of the function, ensuring it covers all possible values of the bill amount.
Being able to read and interpret such definitions is crucial for understanding more complex mathematical models.